Antimagic Labelings of Regular Bipartite Graphs Daniel Cranston dcransto@dimacs.rutgers.edu DIMACS, Rutgers University
Antimagic Labelings Def. magic labeling: an injection from the edges of G to { 1 , 2 , . . . , | E |} such that the sum of the labels incident to each ver- tex is the same Def. a graph is magic if it has an magic labeling
Antimagic Labelings Def. antimagic labeling: an injection from the edges of G to { 1 , 2 , . . . , | E |} such that the sum of the labels incident to each ver- tex is distinct Def. a graph is antimagic if it has an antimagic labeling
Antimagic Labelings Def. antimagic labeling: an injection from the edges of G to { 1 , 2 , . . . , | E |} such that the sum of the labels incident to each ver- tex is distinct Def. a graph is antimagic if it has an antimagic labeling Conj. [Ringel 1990] Every connected graph other than K 2 is antimagic.
Antimagic Labelings Def. antimagic labeling: an injection from the edges of G to { 1 , 2 , . . . , | E |} such that the sum of the labels incident to each ver- tex is distinct Def. a graph is antimagic if it has an antimagic labeling Conj. [Ringel 1990] Every connected graph other than K 2 is antimagic. [Alon et al. 2004] ∃ C s.t. ∀ n if G has n vertices and Thm. δ ( G ) ≥ C log n , then G is antimagic.
Antimagic Labelings Def. antimagic labeling: an injection from the edges of G to { 1 , 2 , . . . , | E |} such that the sum of the labels incident to each ver- tex is distinct Def. a graph is antimagic if it has an antimagic labeling Conj. [Ringel 1990] Every connected graph other than K 2 is antimagic. [Alon et al. 2004] ∃ C s.t. ∀ n if G has n vertices and Thm. δ ( G ) ≥ C log n , then G is antimagic. Thm. [Alon et al. 2004] If ∆( G ) ≥ n − 2, then G is antimagic.
Antimagic Labelings Def. antimagic labeling: an injection from the edges of G to { 1 , 2 , . . . , | E |} such that the sum of the labels incident to each ver- tex is distinct Def. a graph is antimagic if it has an antimagic labeling Conj. [Ringel 1990] Every connected graph other than K 2 is antimagic. [Alon et al. 2004] ∃ C s.t. ∀ n if G has n vertices and Thm. δ ( G ) ≥ C log n , then G is antimagic. Thm. [Alon et al. 2004] If ∆( G ) ≥ n − 2, then G is antimagic. Pf. for ∆( G ) = n − 1. Let d ( v ) = n − 1. Label G − v arbitrarily. Label the final star in order of partial sum.
Antimagic Labelings Def. antimagic labeling: an injection from the edges of G to { 1 , 2 , . . . , | E |} such that the sum of the labels incident to each ver- tex is distinct Def. a graph is antimagic if it has an antimagic labeling Conj. [Ringel 1990] Every connected graph other than K 2 is antimagic. [Alon et al. 2004] ∃ C s.t. ∀ n if G has n vertices and Thm. δ ( G ) ≥ C log n , then G is antimagic. Thm. [Alon et al. 2004] If ∆( G ) ≥ n − 2, then G is antimagic. Pf. for ∆( G ) = n − 1. Let d ( v ) = n − 1. Label G − v arbitrarily. Label the final star in order of partial sum. [Alon et al. 2004] Every complete partite graph other than Thm. K 2 is antimagic.
Antimagic Labelings [Cranston 2007] All k -regular bipartite graphs with k ≥ 2 Thm. are antimagic.
Antimagic Labelings [Cranston 2007] All k -regular bipartite graphs with k ≥ 2 Thm. are antimagic. Prop. Cycles are antimagic.
Antimagic Labelings [Cranston 2007] All k -regular bipartite graphs with k ≥ 2 Thm. are antimagic. Prop. Cycles are antimagic. Pf. 3 5 n − 3 1 n − 1 n 2 n − 2 4 6
Antimagic Labelings [Cranston 2007] All k -regular bipartite graphs with k ≥ 2 Thm. are antimagic. Prop. Cycles are antimagic. Pf. 3 5 n − 3 1 n − 1 n 2 n − 2 4 6 Prop. If G 1 and G 2 are k -regular and antimagic, then so is their disjoint union.
Antimagic Labelings [Cranston 2007] All k -regular bipartite graphs with k ≥ 2 Thm. are antimagic. Prop. Cycles are antimagic. Pf. 3 5 n − 3 1 n − 1 n 2 n − 2 4 6 Prop. If G 1 and G 2 are k -regular and antimagic, then so is their disjoint union. Pf. Increase each label on G 2 by m 1 .
Antimagic Labelings [Cranston 2007] All k -regular bipartite graphs with k ≥ 2 Thm. are antimagic. Prop. Cycles are antimagic. Pf. 3 5 n − 3 1 n − 1 n 2 n − 2 4 6 Prop. If G 1 and G 2 are k -regular and antimagic, then so is their disjoint union. Pf. Increase each label on G 2 by m 1 . Today, I’ll prove the theorem for k ≥ 5 odd.
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 .
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 . ◮ G 1 is (2 l + 2)-regular and resolves conflicts between A and B .
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 . ◮ G 1 is (2 l + 2)-regular and resolves conflicts between A and B . ◮ in G 1 , sums in A equal t and sums in B not equal t (mod3)
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 . ◮ G 1 is (2 l + 2)-regular and resolves conflicts between A and B . ◮ in G 1 , sums in A equal t and sums in B not equal t (mod3) ◮ G 2 is 3-regular and resolves conflicts within A and within B .
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 . ◮ G 1 is (2 l + 2)-regular and resolves conflicts between A and B . ◮ in G 1 , sums in A equal t and sums in B not equal t (mod3) ◮ G 2 is 3-regular and resolves conflicts within A and within B . ◮ in G 2 , sums in A and in B are distinct multiples of 3
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 . ◮ G 1 is (2 l + 2)-regular and resolves conflicts between A and B . ◮ in G 1 , sums in A equal t and sums in B not equal t (mod3) ◮ G 2 is 3-regular and resolves conflicts within A and within B . ◮ in G 2 , sums in A and in B are distinct multiples of 3 ◮ order sums in G 2 in B to match order of sums in G 1 in B
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 . ◮ G 1 is (2 l + 2)-regular and resolves conflicts between A and B . ◮ in G 1 , sums in A equal t and sums in B not equal t (mod3) ◮ G 2 is 3-regular and resolves conflicts within A and within B . ◮ in G 2 , sums in A and in B are distinct multiples of 3 ◮ order sums in G 2 in B to match order of sums in G 1 in B 42 42 42 42 G 1 16 41 41 70
Odd degree at least 5 constructing an antimagic labeling: resolving all potential conflicts Plan for odd degree 2 l + 5 ◮ Decompose G into regular subgraphs G 1 , G 2 . ◮ G 1 is (2 l + 2)-regular and resolves conflicts between A and B . ◮ in G 1 , sums in A equal t and sums in B not equal t (mod3) ◮ G 2 is 3-regular and resolves conflicts within A and within B . ◮ in G 2 , sums in A and in B are distinct multiples of 3 ◮ order sums in G 2 in B to match order of sums in G 1 in B 24+42 30+42 21+42 27+42 G 1 ∪ G 2 21+16 24+41 27+41 30+70
Odd degree at least 5 Let G be bipartite of degree 2 l + 2. Let Lem. t = ( l + 1)(2 ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t (mod3).
Odd degree at least 5 Let G be bipartite of degree 2 l + 2. Let Lem. t = ( l + 1)(2 ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t (mod3). Pf. Partition labels into pairs with sum 2 ln + 1: (0 , 0) and (1 , 2) mod 3. Decompose 2 l -factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3).
Odd degree at least 5 Let G be bipartite of degree 2 l + 2. Let Lem. t = ( l + 1)(2 ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t (mod3). Pf. Partition labels into pairs with sum 2 ln + 1: (0 , 0) and (1 , 2) mod 3. Decompose 2 l -factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3). l 2-factors like this
Odd degree at least 5 Let G be bipartite of degree 2 l + 2. Let Lem. t = ( l + 1)(2 ln + 1). We can label G so the sum at each vertex of A is t and at each vertex of B is not equal to t (mod3). Pf. Partition labels into pairs with sum 2 ln + 1: (0 , 0) and (1 , 2) mod 3. Decompose 2 l -factor into l 2-factors; at each vertex of A use pair of labels, at each vertex of B labels sum to 0(mod3). Remaining 2-factor: at each vertex of A use pair of labels, at each vertex of B labels don’t sum to 0(mod3). 0 0 0 0 0 0 0 0 l 2-factors like this
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